- BEHAVIOR OF μ(g,'T) FOR 'T SMALL 17
PROOF. Suppose that (17.47) is not true. Then there exist s > 0 and a
sequence { TihEN with Ti '\i 0 such that
μ(g,Ti) :::;-f;,
We assume that Ti:::;~· We shall derive a contradiction to (17.46).
Consider the rescaled metrics
1(17.48) 9i ~ -2 g
Ti
(note thatμ (gi, ~) = μ(g,Ti):::; -s). By Proposition 17.24 below, there
exists a corresponding sequence {fihEN of minimizers ofW (g, ' , Ti) = W ( 9i, · , ~)
subject to the constraints(17.49) JM (41rTi)-nl^2 e-fidμ 9 =JM (27r)-n/^2 e-fidμ 9 i = 1.
Then(17.50)Let {xihEN be a sequence of points in M such that
(17.51) Ji( xi) = min fi (x).
xEM
The pointed sequence of Riemannian manifolds {(Mn, gi, Xi)}iEN converges
in the C^00 Cheeger-Gromov sense to Euclidean n-space (JR.n, gJRn, 0). That
is, there exists an exhaustion {Ui}iEN of JR.n by relatively compact open sets
(Ui c ui+l) and embeddings <I>i : ui --+ M such that <I>i (0) =Xi and
(17.52) SJi ~ <I>igi--+ 9JRn
uniformly in C^00 on compact subsets of JR.n.^3
Consider the positive functions
Wi ~ (27r)-n/4 e-Ji/2,which by (17.15) satisfy
(17.53) -2Ll 9 iwi+~R 9 iwi-2Wi logwi-(~ log (27r) + n) Wi = μ (gi, ~) Wi
with the constraint (17.49), i.e.,
(17.54)The contradiction to (17.45) is obtained via the following steps.(^3) See the notes and commentary at the end of this chapter for explicit choices of Ui
and i.